The Heisenberg group has a well studied sub-Riemannian structure. Geometric measure theory in the sub-Riemannian setting is still in development and several fundamental questions are still open. One reason is that sets of finite sub-Riemannian perimeter may have fractal behaviours.

I will present some of the most recent results on minimal surfaces in this setting, in particular: an example of a stable surface that is not area minimizer, and some remarks about contact variations of the area functional.